Understanding Work and Time Problems

Work and time problems are fundamental concepts in quantitative aptitude, frequently appearing in various competitive examinations. These problems involve a direct or inverse relationship between the number of workers, the time taken, and the amount of work completed. The key is to understand how these quantities interact.

Men and Days Relationship

In this specific problem, we are comparing the time taken by two different groups of men to complete the same work. When the amount of work remains constant, the number of men and the number of days required to complete the work are inversely proportional. This means:

• If the number of men increases, the number of days required to complete the work will decrease.
• If the number of men decreases, the number of days required to complete the work will increase.

This inverse relationship is crucial for solving such problems accurately.

Calculating Total Work Done

The total work done is considered constant in such scenarios. We can represent this constant work as the product of the number of men and the number of days they work. Let:

• \(M_1\) = Initial number of men
• \(D_1\) = Initial number of days
• \(M_2\) = New number of men
• \(D_2\) = New number of days

Since the total work remains the same, we can establish the following relationship:

$$M_1 \times D_1 = M_2 \times D_2$$

This formula allows us to find an unknown variable when the other three are known.

Applying Values to the Problem

Let's extract the given information from the question:

Solving for Days

Now, we will substitute these values into our inverse proportion formula:

$$M_1 \times D_1 = M_2 \times D_2$$

Substituting the given values:

$$12 \text{ men} \times 20 \text{ days} = 15 \text{ men} \times D_2 \text{ days}$$

First, calculate the product on the left side, which represents the total work units:

$$12 \times 20 = 240$$

So, the total work is 240 "man-days". Now, we set this equal to the work done by 15 men in \(D_2\) days:

$$240 = 15 \times D_2$$

To find the number of days (\(D_2\)), we need to isolate \(D_2\) by dividing 240 by 15:

$$D_2 = \frac{240}{15}$$

Performing the division:

$$D_2 = 16$$

Therefore, 15 men will complete the same work in 16 days.